# CASA Function: toPara

Converts an algebraic set given in implicit or projected representation to an algebraic set in parametric representation.

### Calling Sequence:

- B := toPara(A)
- B := toPara(A,var)
- B := toPara(A,var,options)

### Parameters:

- A : {algset("impl"),algset("proj"),algset("para")}
- Algebraic set in impl para or projected representation.

- var : name
- A variable name that is used for the parametrization.

- options : list
- A list of options. Where an option is either one of the strings "optimal" (compute a parametrization over an optimal extension field), "realpar" (compute a real parametrization), "check" (plug in the resulting parametrization in the given equations), "points"= <list of simple points on the curve> (a list of points on the curve that could be helpful for the parametrization. The points which are not on the curve will be automatically removed.
- Note, the user should know that supplying simple points may not be a good idea for a non-expert since it may lead to a non-optimal parametrization.

### Result:

- B : algset("para")
- A parametric representation of the algebraic set A.

### Description:

- This function takes an algebraic set and converts it into a parametric representation in affine space. Currently, it may be applied only to one-dimensional algebraic sets (i.e. both plane and space algebraic curves).
- The variables vars will be taken for the parametrization. If no variables are given, the function tries to take the variables from the given algebraic set.

### Examples:

`> ` **A:=mkImplAlgSet([y^2*z^3-x^5],[x,y,z],["basespace"="projective"]);**

`> ` **toPara(A);**

`> ` **A:=mkImplAlgSet([y^4-10*x*y^3+35*x^2*y^2-50*x^3*y+24*x^4+x^3],[x,y]);**

`> ` **toPara(A);**

`> ` **toPara(A,t,["optimal"]);**

`> ` **A:=mkImplAlgSet([x^2-y^3,z+y+x],[x,y,z]);**

`> ` **toPara(A);**

`> ` **A := mkProjAlgSet([[u^2-v^3+v^2],[u+v,u-v,1/v]],[u,v]);**

`> ` **toPara(A);**

### See Also:

[CASA]
[toImpl]
[toProj]
[toPlac]